Nearly Kähler manifold

In mathematics, a nearly Kähler manifold is an almost Hermitian manifold $M$ , with almost complex structure $J$ , such that the (2,1)-tensor $\nabla J$ is skew-symmetric. So,

$(\nabla_X J)X =0 \,$

for every vectorfield $X$ on $M$ .

In particular, a Kähler manifold is nearly Kähler. The converse is not true. The nearly Kähler six-sphere $S 6$ is an example of a nearly Kähler manifold that is not Kähler.^[1] The familiar almost complex structure on the six-sphere is not induced by a complex atlas on $S 6$ .

A nearly Kähler manifold should not be confused with an almost Kähler manifold. An almost Kähler manifold $M$ is an almost Hermitian manifold with a closed Kähler form: $d ω = 0$ . The Kähler form or fundamental 2-form $ω$ is defined by

$\omega(X,Y) = g(JX,Y), \,$

where $g$ is the metric on $M$ . The nearly Kähler six-sphere is an example of a nearly Kähler manifold that is not almost Kähler.

References

^ F. Dillen and L. Verstraelen, ed. Handbook of Differential Geometry, volume II. ISBN 978-0444822406. North Holland.

Categories:

Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

Alfred Gray — For the Commandant of the Marine Corps, see Alfred M. Gray, Jr.. For the 19th century Kansas politician, see Alfred Gray (Kansas politician). Alfred Gray (October 22, 1939 – October 27, 1998) was an American mathematician whose main research… … Wikipedia
Moduli (physics) — In quantum field theory, the term moduli (or more properly moduli fields) is sometimes used to refer to scalar fields whose potential energy function has continuous families of global minima. Such potential functions frequently occur in… … Wikipedia

Academic Dictionaries and Encyclopedias

Nearly Kähler manifold

References

Look at other dictionaries:

Share the article and excerpts

Academic Dictionaries and Encyclopedias

Wikipedia

Nearly Kähler manifold

References

Look at other dictionaries:

Share the article and excerpts

Direct link